Simplify each expression. Write answers without negative exponents. Assume all variables represent nonzero real numbers. (3pq)q²/6p²q⁴

Verified step by step guidance

1. Start by writing the expression clearly: \(\frac{(3pq)q^{2}}{6p^{2}q^{4}}\).
2. Distribute the exponent on the numerator: \((3pq)q^{2} = 3p q \cdot q^{2} = 3p q^{1+2} = 3p q^{3}\).
3. Rewrite the expression with simplified numerator: \(\frac{3p q^{3}}{6 p^{2} q^{4}}\).
4. Separate the fraction into coefficients and variables: \(\frac{3}{6} \times \frac{p}{p^{2}} \times \frac{q^{3}}{q^{4}}\).
5. Simplify each part using exponent rules: \(\frac{3}{6} = \frac{1}{2}\), \(\frac{p}{p^{2}} = p^{1-2} = p^{-1}\), and \(\frac{q^{3}}{q^{4}} = q^{3-4} = q^{-1}\). Then rewrite without negative exponents by moving terms with negative exponents to the denominator.

Key Concepts

Laws of Exponents

The laws of exponents govern how to multiply, divide, and simplify expressions with powers. Key rules include adding exponents when multiplying like bases, subtracting exponents when dividing, and understanding zero and negative exponents. These rules help simplify expressions like (3pq)q² / (6p²q⁴).

Simplifying Algebraic Fractions

Simplifying algebraic fractions involves reducing the numerator and denominator by canceling common factors. This process often requires factoring expressions and applying exponent rules to combine like terms, resulting in a simpler expression without negative exponents.

Writing Expressions Without Negative Exponents

Negative exponents indicate reciprocals, so expressions with negative exponents must be rewritten by moving factors between numerator and denominator. For example, x⁻² becomes 1/x². This ensures the final answer is expressed with only positive exponents, as required.